We use the Chern-Simons (CS) fermion representation of
s=1/2 spin operators to construct topological, long-range magnetically ordered states of interacting two-dimensional (2D) quantum spin models. We show that the fermion-fermion interactions mediated by the dynamic CS flux attachment may give rise to Cooper pairing of the fermions. Specifically, in an
XY model on the honeycomb lattice, this construction leads to a ``CS superconductor'', which belongs to a topologically nontrivial in 2D symmetry class DIII, with particle-hole and time-reversal symmetries. It is shown that in the original spin language, this state corresponds to a symmetry protected topological state, which coexists with a magnetic long-range order. We discuss physical manifestations of the topological character of the corresponding state.
We develop an approach to describe antiferromagnetic magnons on a bipartite lattice supporting the N\'eel state using fractionalized degrees of freedom typically inherent to quantum spin liquids. In particular, we consider a long-range magnetically ordered state of interacting two-dimensional quantum spin-
1/2 XY models using the Chern-Simons (CS) fermion representation of interacting spins. The interaction leads to Cooper instability and pairing of CS fermions, and to CS superconductivity which spontaneously violates the continuous
U(1) symmetry generating a linearly dispersing gapless Nambu-Goldstone mode due to phase fluctuations. We evaluate this mode and show that it is in high-precision agreement with magnons of the corresponding N\'eel antiferromagnet irrespective to the lattice symmetry. Using the fermion formulation of the system with frustration, we show that the competing interactions emerge in the form of long-range interaction vertices mediated by the CS gauge field, which are responsible for restoring the continuous symmetry at sufficiently strong frustration. We identify these new interaction vertices and discuss their implications for unconventional phase transitions. We also apply the proposed theory to a model of anyons that can be tuned continuously from fermions to bosons, and discuss the results.
We investigate pairing instabilities of free electrons caused by the proximity-effect of the chiral spin liquid. We explore a theoretical model where a frustrated spin-half system on a moat-band lattice with degenerate dispersion is immersed in an environment of free fermion bath. Ordering of spins, or condensation of hardcore bosons, is prevented by spontaneous formation of the chiral spin liquid on the moat-band square lattice. We use the spin exchange interaction between spins in the CSL state and the fermionic environment as a starting point to demonstrate, that (i) the Chern-Simons gauge field of the CSL induces a nontrivial effective interaction between fermions; (ii) the fermion interaction further induces a superconducting instability. Modulation of the Chern-Simons flux within the unit cell of the lattice, which is an inherent characteristic of the CSL, gives rise to a d-wave pairing symmetry of the induced superconductor. This microscopic model provides a scenario describing how a quantum spin-liquid state may lead to d-wave superconductivity upon carrier doping. We argue that this effect can be attributed to the pairing interaction induced by the proximity to the quantum spin-liquid.